Some Fine Properties of BV Functions on Wiener Spaces

Luigi Ambrosio; Michele Miranda Jr.; Diego Pallara

Analysis and Geometry in Metric Spaces (2015)

  • Volume: 3, Issue: 1, page 212-230, electronic only
  • ISSN: 2299-3274

Abstract

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In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.

How to cite

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Luigi Ambrosio, Michele Miranda Jr., and Diego Pallara. "Some Fine Properties of BV Functions on Wiener Spaces." Analysis and Geometry in Metric Spaces 3.1 (2015): 212-230, electronic only. <http://eudml.org/doc/271753>.

@article{LuigiAmbrosio2015,
abstract = {In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.},
author = {Luigi Ambrosio, Michele Miranda Jr., Diego Pallara},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Wiener space; functions of bounded variation},
language = {eng},
number = {1},
pages = {212-230, electronic only},
title = {Some Fine Properties of BV Functions on Wiener Spaces},
url = {http://eudml.org/doc/271753},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Luigi Ambrosio
AU - Michele Miranda Jr.
AU - Diego Pallara
TI - Some Fine Properties of BV Functions on Wiener Spaces
JO - Analysis and Geometry in Metric Spaces
PY - 2015
VL - 3
IS - 1
SP - 212
EP - 230, electronic only
AB - In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.
LA - eng
KW - Wiener space; functions of bounded variation
UR - http://eudml.org/doc/271753
ER -

References

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  1. [1] Alberti, C. Mantegazza, A note on the theory of SBV functions, Boll. Un. Mat. Ital. B (7) 11 (1997), n.2, 375–382.  Zbl0877.49001
  2. [2] L. Ambrosio, A. Figalli, Surface measure and convergence of the Ornstein-Uhlenbeck semigroup inWiener spaces, Ann. Fac. Sci. Toulouse Math., 20(2011) 407-438.  Zbl1228.60063
  3. [3] L. Ambrosio, A. Figalli, E. Runa, On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24(2013), 111-122. [WoS] Zbl1282.28002
  4. [4] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, 2000.  Zbl0957.49001
  5. [5] L. Ambrosio, S. Maniglia, M. Miranda Jr, D. Pallara, BV functions in abstract Wiener spaces, J. Funct. Anal., 258(2010), 785– 813.  Zbl1194.46066
  6. [6] L. Ambrosio, M. Miranda Jr, D. Pallara, Sets with finite perimeter in Wiener spaces, perimeter measure and boundary recti- fiability, Discrete Contin. Dyn. Syst., 28(2010), 591–606.  Zbl1196.28023
  7. [7] V. I. Bogachev, Gaussian Measures, American Mathematical Society, Providence R.I., 1998.  
  8. [8] V. I. Bogachev, A.Yu. Pilipenko, A.V. Shaposhnikov, Sobolev functions on infinite-dimensional domains, J.Math. Anal. Appl., 419(2014), 1023–1044.  Zbl1310.46035
  9. [9] V. Caselles, A. Lunardi, M. Miranda Jr, M. Novaga, Perimeter of sublevel sets in infinite dimensional spaces, Adv. Calc. Var., 5(2012), 59–76. [WoS] Zbl1257.49006
  10. [10] P. Celada, A. Lunardi, Traces of Sobolev functions on regular surfaces in infinite dimensions, J. Funct. Anal., 266(2014), 1948–1987. [WoS] Zbl1308.46042
  11. [11] E. De Giorgi, L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8) 82(1988), n.2, 199–210, English translation in: Ennio De Giorgi: Selected Papers, (L. Ambrosio, G. DalMaso, M. Forti, M. Miranda, S. Spagnolo eds.) Springer, 2006, 686–696.  
  12. [12] N. Dunford, J.T. Schwartz, Linear operators Part I: General theory, Wiley, 1958.  Zbl0084.10402
  13. [13] D. Feyel, A. de la Pradelle, Hausdorff measures on the Wiener space, Potential Anal. 1(1992), 177-189.  Zbl1081.28500
  14. [14] M. Fukushima, BV functions and distorted Ornstein-Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal., 174(2000), 227-249.  Zbl0978.60088
  15. [15] M. Fukushima, M. Hino, On the space of BV functions and a Related Stochastic Calculus in Infinite Dimensions, J. Funct. Anal., 183(2001), 245-268.  Zbl0993.60049
  16. [16] L. Gross, Abstract Wiener spaces, in: Proc. Fifth Berkeley Symp. Math. Stat. Probab. (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, p. 31-42, Univ. California Press, Berkeley.  
  17. [17] M. Hino, Sets of finite perimeter and the Hausdorff–Gauss measure on the Wiener space, J. Funct. Anal., 258(2010), 1656– 1681. [WoS] Zbl1196.46029
  18. [18] M. Hino, H. Uchida, Reflecting Ornstein-Uhlenbeck processes on pinned path spaces, Proceedings of RIMS Workshop on Stochastic Analysis and Applications, 111–128, RIMS Kokyuroku Bessatsu, B6, Kyoto, 2008.  Zbl1143.60050
  19. [19] M. Ledoux, Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space, Bull. Sci. Math., 118(1994), 485–510.  Zbl0841.49024
  20. [20] P. Malliavin, Stochastic analysis, Grundlehren der Mathematischen Wissenschaften 313, Springer, 1997.  
  21. [21] M.Miranda Jr, M. Novaga, D. Pallara, An introduction to BV functions in Wiener spaces, Advanced Studies in Pure Mathematics, 67, 245–293, Tokyo 2015.  
  22. [22] M. Miranda Jr, D. Pallara, F. Paronetto, M. Preunkert, Short–time heat flow and functions of bounded variation in RN, Ann. Fac. Sci. Toulouse, XVI(2007), 125–145.  Zbl1142.35445
  23. [23] R. O’Donnell, Analysis of Boolean functions, Cambridge University Press, 2014.  
  24. [24] M. Röckner, R.C. Zhu, X.C. Zhu, The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple, Ann. Probab., 40(2012), 1759-1794. [WoS] Zbl1267.60074
  25. [25] D. Trevisan:, BV-regularity for the Malliavin derivative of the maximum of the Wiener process, Electron. Commun. Probab., 18(2013), no.29. [WoS][Crossref] Zbl1309.60054
  26. [26] D. Trevisan, Lagrangian flows driven by BV fields in Wiener spaces, Probab. Theory Related Fields, DOI 10.1007/s00440- 014-0589-1. [Crossref] Zbl1329.35109
  27. [27] A.I. Vol’pert, S.I. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, Martinus Nijhoff Publishers, Dordrecht, NL, 1985.  
  28. [28] L. Zambotti, Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Related Fields, 123(2002) 579-600.  Zbl1009.60047

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